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abelian group attaching map basepoint cells chain complex Chern classes closed form cochain cocycle coefficients commutative diagram constant map construct Corollary CW complex D-vector definition degree denote deRham theorem differential algebra differential forms dual element EP,q exercise f is homotopic finite follows functor gives H(dv Hence Hirsch extension homology homomorphism homotopy class homotopy equivalence homotopy groups homotopy lifting homotopy theory homotopy type Hurewicz theorem induces an isomorphism isomorphism on cohomology k-invariant l-minimal Lemma Let f Lie algebra long exact sequence manifold map cp map f minimal model morphism n-cell nilpotent nonzero obstruction theory obstruction to extending P.L. forms pair polynomial principal fibration Proof prove rational homotopy Serre spectral sequence simplex simplicial complex simply connected space singular singular homology sphere Suppose t,dt topology triangulation vanishes vector bundle vector space Whitehead product X U A X;tt zero